Permeability Upscaling from Core-Scale to Log-Scale
Mohammed Abdul Ameer-Basra University For Oil and Gas
data<-read.csv("D:/karpur.csv",header=T)
summary(data)## depth caliper ind.deep ind.med
## Min. :5667 Min. :8.487 Min. : 6.532 Min. : 9.386
## 1st Qu.:5769 1st Qu.:8.556 1st Qu.: 28.799 1st Qu.: 27.892
## Median :5872 Median :8.588 Median :217.849 Median :254.383
## Mean :5873 Mean :8.622 Mean :275.357 Mean :273.357
## 3rd Qu.:5977 3rd Qu.:8.686 3rd Qu.:566.793 3rd Qu.:544.232
## Max. :6083 Max. :8.886 Max. :769.484 Max. :746.028
## gamma phi.N R.deep R.med
## Min. : 16.74 Min. :0.0150 Min. : 1.300 Min. : 1.340
## 1st Qu.: 40.89 1st Qu.:0.2030 1st Qu.: 1.764 1st Qu.: 1.837
## Median : 51.37 Median :0.2450 Median : 4.590 Median : 3.931
## Mean : 53.42 Mean :0.2213 Mean : 24.501 Mean : 21.196
## 3rd Qu.: 62.37 3rd Qu.:0.2640 3rd Qu.: 34.724 3rd Qu.: 35.853
## Max. :112.40 Max. :0.4100 Max. :153.085 Max. :106.542
## SP density.corr density phi.core
## Min. :-73.95 Min. :-0.067000 Min. :1.758 Min. :15.70
## 1st Qu.:-42.01 1st Qu.:-0.016000 1st Qu.:2.023 1st Qu.:23.90
## Median :-32.25 Median :-0.007000 Median :2.099 Median :27.60
## Mean :-30.98 Mean :-0.008883 Mean :2.102 Mean :26.93
## 3rd Qu.:-19.48 3rd Qu.: 0.002000 3rd Qu.:2.181 3rd Qu.:30.70
## Max. : 25.13 Max. : 0.089000 Max. :2.387 Max. :36.30
## k.core Facies
## Min. : 0.42 Length:819
## 1st Qu.: 657.33 Class :character
## Median : 1591.22 Mode :character
## Mean : 2251.91
## 3rd Qu.: 3046.82
## Max. :15600.00First step is to see if there is a direct relationship between ’Φ.Core” and “Φ.Log”
par(mfrow=c(1,1))
model1 <- lm(data$phi.core ~ data$phi.N)
coef(model1)## (Intercept) data$phi.N
## 30.96232 -18.20696plot(data$phi.N,data$phi.core, xlab = 'Φ.Log',ylab='Φ.Core (%)')
abline(model1, lwd=2, col='blue' )
summary(model1)##
## Call:
## lm(formula = data$phi.core ~ data$phi.N)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.5237 -3.0779 0.9432 3.3563 10.4025
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.962 0.485 63.846 <2e-16 ***
## data$phi.N -18.207 2.080 -8.753 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.368 on 817 degrees of freedom
## Multiple R-squared: 0.08573, Adjusted R-squared: 0.08462
## F-statistic: 76.61 on 1 and 817 DF, p-value: < 2.2e-16model1 <- lm((data$phi.core/100) ~., data = data)
summary(model1)##
## Call:
## lm(formula = (data$phi.core/100) ~ ., data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.099073 -0.008962 0.000796 0.008740 0.090767
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.613e+00 2.577e-01 6.261 6.24e-10 ***
## depth -3.948e-05 2.686e-05 -1.470 0.142023
## caliper -8.627e-02 1.484e-02 -5.812 8.90e-09 ***
## ind.deep 5.081e-05 3.492e-05 1.455 0.146104
## ind.med -3.357e-05 3.839e-05 -0.874 0.382160
## gamma 3.993e-05 9.246e-05 0.432 0.665963
## phi.N 1.368e-01 2.141e-02 6.392 2.78e-10 ***
## R.deep -1.528e-04 9.406e-05 -1.624 0.104683
## R.med 1.954e-04 1.377e-04 1.419 0.156296
## SP -4.974e-06 4.686e-05 -0.106 0.915504
## density.corr -1.546e-01 7.125e-02 -2.170 0.030298 *
## density -1.865e-01 1.618e-02 -11.531 < 2e-16 ***
## k.core 4.240e-06 5.037e-07 8.418 < 2e-16 ***
## FaciesF10 -4.419e-02 5.120e-03 -8.631 < 2e-16 ***
## FaciesF2 -7.517e-02 8.238e-03 -9.126 < 2e-16 ***
## FaciesF3 -2.873e-02 4.868e-03 -5.900 5.35e-09 ***
## FaciesF5 -3.388e-02 4.964e-03 -6.824 1.75e-11 ***
## FaciesF7 -5.684e-02 8.290e-03 -6.857 1.41e-11 ***
## FaciesF8 -1.282e-02 5.849e-03 -2.193 0.028630 *
## FaciesF9 -2.453e-02 6.515e-03 -3.765 0.000179 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01874 on 799 degrees of freedom
## Multiple R-squared: 0.8354, Adjusted R-squared: 0.8315
## F-statistic: 213.4 on 19 and 799 DF, p-value: < 2.2e-16Now we see that R square and Adjusted R square became 0.8354, 0.8315 respectively which indcate a good relationship,therefore, we have 13 parameters influence in the relationship between ’Φ.Core” and “Φ.Log”. So lets write the multiple linear regression to find “Φ.CoreL”
model1 <- lm((data$phi.core/100) ~caliper+phi.N+density.corr+density+k.core+Facies, data = data)
summary(model1)##
## Call:
## lm(formula = (data$phi.core/100) ~ caliper + phi.N + density.corr +
## density + k.core + Facies, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.099953 -0.009649 -0.000134 0.009053 0.090382
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.231e+00 8.952e-02 13.746 < 2e-16 ***
## caliper -6.540e-02 9.916e-03 -6.595 7.68e-11 ***
## phi.N 1.579e-01 1.770e-02 8.918 < 2e-16 ***
## density.corr -1.667e-01 7.013e-02 -2.376 0.0177 *
## density -1.974e-01 1.540e-02 -12.819 < 2e-16 ***
## k.core 3.996e-06 4.363e-07 9.158 < 2e-16 ***
## FaciesF10 -4.517e-02 5.037e-03 -8.967 < 2e-16 ***
## FaciesF2 -7.914e-02 8.051e-03 -9.829 < 2e-16 ***
## FaciesF3 -3.000e-02 4.684e-03 -6.404 2.57e-10 ***
## FaciesF5 -3.794e-02 3.788e-03 -10.018 < 2e-16 ***
## FaciesF7 -6.198e-02 7.222e-03 -8.582 < 2e-16 ***
## FaciesF8 -1.863e-02 3.765e-03 -4.948 9.11e-07 ***
## FaciesF9 -2.693e-02 3.821e-03 -7.047 3.92e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0189 on 806 degrees of freedom
## Multiple R-squared: 0.8312, Adjusted R-squared: 0.8287
## F-statistic: 330.7 on 12 and 806 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(model1,lwd=3)
Predict and plot the “Φ.CoreL”
Φ.CoreL<-predict(model1,data)
plot(Φ.CoreL * 100,xlab='Number of Samples',ylab='Φ.CoreL')
#cbind(data$k.core,Φ.CoreL)Now we have “Φ.CoreL” let’s make relationship between it and K.core by using simple linear regression to find “K.coreL”
model2 <- lm(data$k.core ~ Φ.CoreL)
plot(Φ.CoreL,data$k.core,xlab ='Φ.CoreL',ylab='K.core')
abline(model2, lwd=3, col='red' )
model3 <- lm(data$k.core ~., data = data)
summary(model3)##
## Call:
## lm(formula = data$k.core ~ ., data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5585.6 -568.9 49.2 476.5 8928.4
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.783e+04 1.760e+04 -3.853 0.000126 ***
## depth 8.544e+00 1.785e+00 4.786 2.02e-06 ***
## caliper 1.413e+03 1.019e+03 1.387 0.165789
## ind.deep -2.418e-01 2.354e+00 -0.103 0.918220
## ind.med 1.224e+00 2.585e+00 0.473 0.636062
## gamma -4.583e+01 6.010e+00 -7.626 6.88e-14 ***
## phi.N -2.010e+03 1.476e+03 -1.362 0.173540
## R.deep -2.344e+01 6.288e+00 -3.727 0.000207 ***
## R.med 5.643e+01 9.065e+00 6.225 7.76e-10 ***
## SP -7.125e+00 3.145e+00 -2.266 0.023736 *
## density.corr -2.567e+03 4.809e+03 -0.534 0.593602
## density 2.319e+03 1.173e+03 1.976 0.048458 *
## phi.core 1.921e+02 2.282e+01 8.418 < 2e-16 ***
## FaciesF10 8.921e+02 3.590e+02 2.485 0.013157 *
## FaciesF2 9.243e+02 5.818e+02 1.589 0.112514
## FaciesF3 4.393e+02 3.344e+02 1.313 0.189394
## FaciesF5 7.411e+02 3.428e+02 2.162 0.030908 *
## FaciesF7 -4.152e+01 5.742e+02 -0.072 0.942377
## FaciesF8 -1.179e+03 3.927e+02 -3.002 0.002770 **
## FaciesF9 -2.969e+03 4.298e+02 -6.908 1.00e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1262 on 799 degrees of freedom
## Multiple R-squared: 0.6889, Adjusted R-squared: 0.6815
## F-statistic: 93.12 on 19 and 799 DF, p-value: < 2.2e-16model3.red <- lm(data$k.core~Φ.CoreL+depth+gamma+R.deep+R.med+SP+density+phi.core+Facies, data =data)
coef(model3.red)## (Intercept) Φ.CoreL depth gamma R.deep
## -41763.397880 116464.432647 -5.431412 -77.510493 -16.847085
## R.med SP density phi.core FaciesF10
## 19.043411 -2.609740 22033.559000 13.194473 5237.387181
## FaciesF2 FaciesF3 FaciesF5 FaciesF7 FaciesF8
## 8244.241790 3419.732335 3647.957858 4752.532284 -775.751582
## FaciesF9
## 876.817847par(mfrow=c(2,2))
plot(model3.red,lwd=3)
K.coreL <- predict(model3.red,data)
plot(K.coreL,xlab='Number of Samples',ylab='K.coreL (md)')
AdjR.sq2 <- 1-sum((K.coreL - data$k.core)^2)/sum((data$k.core-mean(data$k.core))^2)
AdjR.sq2## [1] 0.8219384rmse.model3 <- sqrt(sum((K.coreL - data$k.core)^2)/nrow(data))
rmse.model3## [1] 942.7917Add our new data “K.coreL” to the file karpur.csv
karpur2<-cbind(data,K.coreL)
write.csv(karpur2,"karpur2.csv")head(karpur2)## depth caliper ind.deep ind.med gamma phi.N R.deep R.med SP
## 1 5667.0 8.685 618.005 569.781 98.823 0.410 1.618 1.755 -56.587
## 2 5667.5 8.686 497.547 419.494 90.640 0.307 2.010 2.384 -61.916
## 3 5668.0 8.686 384.935 300.155 78.087 0.203 2.598 3.332 -55.861
## 4 5668.5 8.686 278.324 205.224 66.232 0.119 3.593 4.873 -41.860
## 5 5669.0 8.686 183.743 131.155 59.807 0.069 5.442 7.625 -34.934
## 6 5669.5 8.686 109.512 75.633 57.109 0.048 9.131 13.222 -39.769
## density.corr density phi.core k.core Facies K.coreL
## 1 -0.033 2.205 33.9000 2442.590 F1 4757.604
## 2 -0.067 2.040 33.4131 3006.989 F1 4581.700
## 3 -0.064 1.888 33.1000 3370.000 F1 3884.732
## 4 -0.053 1.794 34.9000 2270.000 F1 2621.163
## 5 -0.054 1.758 35.0644 2530.758 F1 2377.844
## 6 -0.058 1.759 35.3152 2928.314 F1 2520.201Now lts’s plot our final work
par(mfrow=(c(1,5)))
plot(data$phi.core,data$depth,ylim =rev(c(5667,6083)), xlim=c(40,10)
,type="l",col="blue", lwd=2, xlab='Φ.Core' ,ylab='depth (m)')
plot(data$phi.N * 100,data$depth,ylim =rev(c(5667,6083)), xlim=c(40,10)
,type="l",col="darkred", lwd=2, xlab='Φ.Log' ,ylab='depth (m)')
plot(Φ.CoreL * 100,data$depth,ylim =rev(c(5667,6083)), xlim=c(40,10)
,type="l",col="yellow", lwd=2, xlab='Φ.CoreL' ,ylab='depth (m)')
plot(data$k.core,data$depth,ylim =rev(c(5667,6083)), xlim=c(0.42,15600.00) ,type="l",col="red", lwd=2, xlab='K.core (md)' ,ylab='depth (m)')
plot(K.coreL,data$depth,ylim =rev(c(5667,6083)), xlim=c(0.42,15600.00),type="l",col="green", lwd=2, xlab='K.coreL (md)' ,ylab='depth (m)')
par(mfrow=c(1,1))
boxplot(data$depth~data$Facies,col="300" ,xlab='Facies',
ylab='Depth (m)',cex=1.5, cex.lab=1.5, cex.axis=1.2,col.axis="100",col.lab="darkgreen")
grid(nx= NULL, ny= NULL, lty=3,col="red",lwd=2)